Chi-Square Calculator

The chi-square statistic for an experiment with k
possible outcomes, performed n times, in which
Y1, Y2,… Yk are the number of
experiments which resulted in each possible outcome, with
probabilities of each outcome p1,
p2,… pk is:

X² will be larger to the extent that the
observed results diverge from those expected by chance. The
probability Q that a X² value calculated
for an experiment with d degrees of freedom (where
d=k−1, one less the number of possible outcomes) is
due to chance is:

Where Γ is the generalisation of the factorial
function to real and complex arguments:

Unfortunately, there is no closed form solution for Q, so it
must be evaluated numerically. This page allows you to calculate the
probability of chance occurrence of a given X² for
an experiment with d degrees of freedom, or to calculate
X² given d and the probability
Q. All calculations are performed with six decimal places of
accuracy; the maximum X² accepted is thus 99999.

Note that the probability calculated from the X²
is an approximation which is valid only for large
values of n, and is therefore only meaningful when
calculated from a large number of independent experiments.

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Calculate probability from X² and d

One of the most common chi-square calculations is determining,
given the measured X² value for a set of
experiments with a degree of freedom d, the probability
of the result being due to chance. Enter the X²
and d values in the boxes below, press the Calculate
button, and the probability will appear in the Q box.

Given X²= and
d=

The chance probability, Q, is:

Calculate X² from probability Q and d

To determine the chi-square value indicating a probability Q
of non-chance occurrence for an experiment with d degrees
of freedom, enter Q and d in the boxes below and
press Calculate.